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dc.contributor.authorLe Donne, Enrico
dc.contributor.authorLi, Sean
dc.contributor.authorRajala, Tapio
dc.date.accessioned2019-03-18T11:24:19Z
dc.date.available2019-03-18T11:24:19Z
dc.date.issued2017
dc.identifier.citationLe Donne, E., Li, S., & Rajala, T. (2017). Ahlfors-regular distances on the Heisenberg group without biLipschitz pieces. <i>Proceedings of the London Mathematical Society</i>, <i>115</i>(2), 348-380. <a href="https://doi.org/10.1112/plms.12044" target="_blank">https://doi.org/10.1112/plms.12044</a>
dc.identifier.otherCONVID_26996337
dc.identifier.otherTUTKAID_73741
dc.identifier.urihttps://jyx.jyu.fi/handle/123456789/63198
dc.description.abstractWe show that the Heisenberg group is not minimal in looking down. This answers Problem 11.15 in Fractured fractals and broken dreams by David and Semmes, or equivalently, Question 22 and hence also Question 24 in Thirty-three yes or no questions about mappings, measures, and metrics by Heinonen and Semmes. The non-minimality of the Heisenberg group is shown by giving an example of an Ahlfors 4-regular metric space X having big pieces of itself such that no Lipschitz map from a subset of X to the Heisenberg group has image with positive measure, and by providing a Lipschitz map from the Heisenberg group to the space X having as image the whole X. As part of proving the above result we define a new distance on the Heisenberg group that is bounded by the Carnot-Carath´eodory distance, that preserves the Ahlfors-regularity, and such that the Carnot-Carath´eodory distance and the new distance are biLipschitz equivalent on no set of positive measure. This construction works more generally in any Ahlfors-regular metric space where one can make suitable shortcuts. Such spaces include for example all snowflaked Ahlfors-regular metric spaces. With the same techniques we also provide an example of a leftinvariant distance on the Heisenberg group biLipschitz to the Carnot-Carath´eodory distance for which no blow-up admits nontrivial dilations.fi
dc.format.mimetypeapplication/pdf
dc.language.isoeng
dc.publisherOxford University Press; London Mathematical Society
dc.relation.ispartofseriesProceedings of the London Mathematical Society
dc.rightsIn Copyright
dc.subject.otherHeisenberg group
dc.subject.otherAhlfors-regular distances
dc.titleAhlfors-regular distances on the Heisenberg group without biLipschitz pieces
dc.typearticle
dc.identifier.urnURN:NBN:fi:jyu-201903121830
dc.contributor.laitosMatematiikan ja tilastotieteen laitosfi
dc.contributor.laitosDepartment of Mathematics and Statisticsen
dc.contributor.oppiaineMatematiikkafi
dc.contributor.oppiaineMathematicsen
dc.type.urihttp://purl.org/eprint/type/JournalArticle
dc.date.updated2019-03-12T13:15:26Z
dc.description.reviewstatuspeerReviewed
dc.format.pagerange348-380
dc.relation.issn0024-6115
dc.relation.numberinseries2
dc.relation.volume115
dc.type.versionacceptedVersion
dc.rights.copyright© 2017 London Mathematical Society
dc.rights.accesslevelopenAccessfi
dc.relation.grantnumber274372
dc.subject.ysomatematiikka
dc.format.contentfulltext
jyx.subject.urihttp://www.yso.fi/onto/yso/p3160
dc.rights.urlhttp://rightsstatements.org/page/InC/1.0/?language=en
dc.relation.doi10.1112/plms.12044
dc.relation.funderSuomen Akatemiafi
dc.relation.funderAcademy of Finlanden
jyx.fundingprogramAkatemiatutkijan tehtävä, SAfi
jyx.fundingprogramResearch post as Academy Research Fellow, AoFen
jyx.fundinginformationSean Li is supported by NSF postdoctoral fellowship DMS‐1303910. Tapio Rajala acknowledges the support of the Academy of Finland project no. 274372.


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